Linear orderings of combinatorial cubes

Abstract

We show that, for every linear ordering of $[2{]}^n$, there is a large subcube on which the ordering is lexicographic. We use this to deduce that every long sequence contains a long monotone subsequence supported on an affine cube. More generally, we prove an analogous result for linear orderings of $[k{]}^n$. We show that, for every such ordering, there is a large subcube on which the ordering agrees with one of approximately $\frac{(k-1)!}{2(\ln 2{)}^k}$ orderings.

9 pages, 2 figures